This invention relates to a method for generating an input to a system to minimize unwanted dynamics.
Control of machines that exhibit unwanted dynamics such as vibration becomes important when designers attempt to push the state of the art with faster, lighter machines. Three steps are necessary for the control of a flexible plant. First, a good model of the plant must exist. Second, a good controller must be designed. Third, inputs to the controller must be constructed using knowledge of the system dynamic response. There is a great deal of literature pertaining to modeling and control but little dealing with the shaping of system inputs. The numbers in brackets below refer to references listed in the Appendix, the teachings of which are hereby incorporated by reference.
Command shaping involves preshaping either actuator commands or setpoints so that unwanted dynamics are reduced. This aspect of control is often ignored because it is mistakenly considered to be useful only for open loop systems. However, if the input shaping accounts for the dynamic characteristics of the closed loop plant, then shaped input commands can be given to the closed loop plant as well. Thus, any of the preshaping techniques may be readily used as a closed loop technique. [11] [16]
The earliest form of command preshaping was the use of high-speed cam profiles as motion templates. These input shapes were generated so as to be continuous throughout one cycle (i.e. the cycloidal cam profile). Their smoothness (continuous derivatives) reduces unwanted dynamics by not putting high frequency inputs into the system [12]; however, these profiles have limited success.
Another early form of setpoint shaping was the use of posicast control by O. J. M. Smith [16]. This technique involves breaking a step of a certain magnitude into two smaller steps, one of which is delayed in time. This results in a response with a reduced settling time. In effect, superposition of the responses leads to vibration cancellation. However, this is not generally used because of problems with robustness. The system that is to be commanded must have only one resonance, be known exactly, and be very linear for this technique to work.
Optimal control approaches have been used to generate input profiles for commanding vibratory systems. Junkins, Turner, Chun, and Juang have made considerable progress toward practical solutions of the optimal control formulation for flexible systems. [18 ] [24] [25] Typically, a penalty function is selected (for example integral squared error plus some control penalty). The resulting "optimal" trajectory is obtained in the form of the solution to the system equations (a model). This input is then given to the system.
Farrenkopf [5] and Swigert [17] demonstrated that velocity and torque shaping can be implemented on systems which modally decompose into second order harmonic oscillators. They showed that inputs in the form of the solutions for the decoupled modes can be added so as not to excite vibration while moving the system. Their technique solves for parameters in a template function, therefore, inputs are limited to the form of the template. These parameters that define the control input are obtained by minimizing some cost function using an optimal formulation. The drawback of this approach is that the inputs are difficult to compute and they must be calculated for each move of the system.
Gupta [22], and Junkins and Turner [18] also included some frequency shaping terms in the optimal formulation. The derivative of the control input is included in the penalty function so that as with cam profiles, the resulting functions are smooth.
Several papers also address the closed loop "optimal" feedback gains which are used in conjunction with the "optimal" open-loop input. [18][24][25]
There are four drawbacks to these "optimal" approaches. First, computation is difficult. Each motion of the system requires recomputation of the control. Though the papers cited above have made major advances toward simplifying this step, it continues to be extremely difficult or impossible to solve for complex systems.
Second, the penalty function does not explicitly include a direct measure of the unwanted dynamics (often vibration). Tracking error is used in the penalty function, therefore, all forms of error are essentially lumped together--the issue of unwanted dynamics is not addressed directly. One side effect is that these approaches penalize residual vibration but allow the system to vibrate during the move. This leads to a lack of robustness under system uncertainties. Removing vibrational energy from a system is difficult especially under conditions of system uncertainty. Techniques that start a move, allowing the system to vibrate and then expect to remove that vibration later in the move lack robustness to slight parameter variations. In addition, vibration is undesirable during a move as well as at the end.
Third, the solutions are limited to the domain of continuous functions. This is an arbitrary constraint which enables the solution of the problem.
Fourth, optimal input strategies depend on move time for how well they work. Different moves will have different vibration excitation levels.
Another technique is based on the concept of the computed torque approach. The system is first modeled in detail. This model is then inverted--the desired output trajectory is specified and the required input needed to generate that trajectory is computed. For linear systems, this might involve dividing the frequency spectrum of the trajectory by the transfer function of the system, thus obtaining the frequency spectrum of the input. For non-linear systems this technique involves inverting the equations for the model. [2]
Techniques that invert the plant have four problems. First, a trajectory must be selected. If the trajectory is impossible to follow, the plant inversion fails to give a usable result. Often a poor trajectory is selected to guarantee that the system can follow it, thus defeating the purpose of the input [2]. Second, a detailed model of the system is required. This is a difficult step for machines which are not extremely simple. Third, the plant inversion is not robust to variations in the system parameters because no robustness criterion has been included in the calculation. Fourth, this technique results in large move time penalties because the plant inversion process results in an acausal input (an input which exists before zero time). In order to use this input, it must be shifted in time thus increasing the move time.
Another approach to command shaping is the work of Meckl and Seering [7] [8] [9] [10] [11]. They investigated several forms of feedforward command shaping. One approach they examined is the construction of input functions from either ramped sinusoids or versine functions. This approach involves adding up harmonics of one of these template functions. If all harmonics were included, the input would be a time optimal rectangular (bang-bang) input function. The harmonics that have significant spectral energy at the natural frequencies of the system are discarded. The resulting input which is given to the system approaches the rectangular shape, but does not excite the resonances.
Aspinwall [1] proposed a similar approach which involves creating input functions by adding harmonics of a sine series. The coefficients of the series are chosen to minimize the frequency content of the input over a band of frequencies. Unlike Meckl, the coefficients were not selected to make the sine series approach a rectangular function, therefore, a large time penalty was incurred.
Wang, Hsia, and Wiederrich [21] proposed yet another approach for creating a command input that moves a flexible system while reducing the residual vibrations. They modeled the system in software and designed a PID controller for the plant that gave a desired response. They then examined the actual input that the controller gave to the software plant and used this for the real system. Next, they refined this input (the reference) with an iteration scheme that adds the error signal to the reference in order to get better tracking of the trajectory. This technique requires accurate modeling of the system and is not robust to parameter uncertainty. In addition, this technique has the implicit assumption that a good response can be achieved with a PID controller. In fact, systems with flexibility cannot be given sufficient damping and a reasonable response time simply by adding a PID controller.
Often a notch filter is proposed for input signal conditioning. This approach gives poor results for several reasons. First, a causal (real time) filter distorts the phase of the resulting signal. This effect is aggravated by lengthening the filter sequence of digital filters or by increasing the order of analog or recursive filters. Therefore, efforts to improve the frequency characteristics of a filter result in increased phase distortion. Also, penalties, such as filter ringing or long move times often result from trying to improve the filter's frequency characteristics.
Singer and Seering [14] investigated an alternative approach of shaping a time optimal input by acausally filtering out the frequency components near the resonances. This has an advantage over notch filtering in that phase distortion and ringing no longer pose a problem. The drawbacks of this approach [14] are the tradeoffs that must be made between fidelity in frequency and reduction of the move time.